Functorial characterization of morphisms of schemes

Question

This question is akin in spirit to this one:

Functorial characterization of open subschemes?

In the above MO question, a "functorial" characterization is given for closed immersions and open immersions. I am wondering if there are similar characterizations for concepts such as universally closed, separated, proper, projective etc.? Or maybe there are characterizations in a different flavor? In particular, I am wondering if we work over functors (i.e., generalization of schemes), are there things like separated/proper/etc morphism between functors?

Any reference or comments would be appreciated!

Answer 1

Check out the paper of Kontsevich-Rosenberg Noncommutative spaces and Noncommutative Grassmannian and related constructions. You will get what you want. i.e. the definition of properness and separatedness of presheaves(as functors, taken as "space") and morphism between presheaves(natural transformations).

Notice that these definitions are general treatment for algebraic geometry in functorial point of view,nothing to do with noncommutative.

Definition for separated morphism and separated presheaves

Let $X$ and $Y$ be presheaves of sets on a category $A$(in particular, $CRings^{op}$). We call a morphism $X\rightarrow Y$ separated if the canonical morphism

$X\rightarrow X\times _{Y}X$ is closed immersion We say a presheaf $X$ on $A$ is separated if the diagonal morphism: $X\rightarrow X\times X$ is closed immersion

Definition for strict monomorphism and closed immersion

For a morphism $f$: $Y\rightarrow X$ of a category $A$, denote by $\Lambda _{f}$ the class of all pairs of morphisms

$u_{1}$,$u_{2}$:$X\Rightarrow V$ equalizing $f$, then $f$ is called a strict monomorphism if any morphism $g$: $Z\rightarrow X$ such that $\Lambda_{f}\subseteq \Lambda_{g}$ has a unique decomposition: $g=f\cdot g'$

Now we have come to the definition of closed immersion: Let $F,G$ be presheaves of sets on $A$. A morphism $F\rightarrow G$ a closed immersion if it is representable by a strict monomorphism.

Example

Let $A$ be the category $CAff/k$ of commutative affine schemes over $Spec(k)$, then strict monomorphisms are exactly closed immersion(classcial sense)of affine schemes. Let $X,Y$ be arbitrary schemes identified with the correspondence sheaves of sets on the category $CAff/k$. Then a morphism $X\rightarrow Y$ is a closed immersion iff it is a closed immersion in classical sense(Hartshorne or EGA)

Definition for proper morphism just follows the classical definition: i.e. universal closed and separated. You can also find the definition of universal closed morphism in functorial flavor in the paper I mentioned.

Answer 2

Another point of view

if you follow the page you quote: Functorial characterization of open subschemes. There are also correspondence notions for separatedness and properness and so on. Let me elaborate a bit.

What you do is to identify a commutative scheme $X$ with $Qcoh_{X}$(Gabriel-Rosenberg reconstruction theorem). Let$f_{*}$=$F$

:$Qcoh_{X}\rightarrow Qcoh_{Y}$ (Assume $X,Y$ are quasi compact and quasi separated).

Affineness $F$ is affine if $f_{*}$ is conservative(faithful in abelian case),having left adjoint functor

$f^{*}$ and having right adjoint functor $f^{!}$

closed immersion

Let $C_{X}=Qcoh_{X}$ and $C_{U}=Qcoh_{U}$.(Suppose they are abelian categories). Then $C_{U}\rightarrow C_{X}$ ($u_{*}$) is closed immersion if ($u_{*}$) is an categorical equivalence of:

$C_{U}$ and full topologizing subcategory $C_{V}$ of $C_{X}$(topologizing subcategory is full subcategory which is closed under finite direct sum and subquotient taken in $C_{X}$)

thickennings

We call a closed immersion $U\rightarrow T$ a thickenning, if the smallest saturated multiplicative system in $HomC_{T}$ containing $(u*)(HomC_{U})$ coincides with $Hom(C_{T})$

Formally smooth,formally unramified,formally etale

I will talk about these notions later. They are defined via thickennings.

separatedness and properness

Once you have the definition of closed immersion given above, then the definition of separatedness is free(follows the same pattern as EGA) Properness is similar. I will formulated later.

Notice

The reason to identify space with category of quasi coherent sheaves on it is mainly for noncommutative algebraic geometry. What I wrote here is trivial case of this consideration because we can drop the categorical language in commutative case.

Functor point of view and categorical point of view are not equivalent in general